1. Mixture & Multilevel Modeling
Shaunna Clark & Ryne Estabrook
NIDA Workshop – October 19, 2010

2. Outline Mixture Models Multilevel Models What is mixture modeling?
Growth Mixture Model
Open Mx
Genetic Mixture Models
Other Longitudinal Mixture Models
Multilevel Models
What is multilevel data?
Multilevel regression model

3. Homogeneity Vs. Heterogeneity
Most models assume homogeneity
i.e. Individuals in a sample all follow the same model
What have seen so far today
But not always the case
Ex: Sex, Age, Alcohol Use Trajectories

4. What is Mixture Modeling
Used to model unobserved heterogeneity in a population by identifying different groups of individuals, called latent classes, that have similar observed response patterns (cross-sectional) or growth trajectory.
Ex: Fish in a stream
Task is to examine length of fish in a stream
Two cohorts of fish – c=0 cohort of younger fish
c = 1 cohort of older fish
Two cohorts mix (hence mixture) to form the overall population
Used to model unobserved heterogeneity by identifying different subgroups of individuals
Ex: IQ, Religiosity

5. Growth Mixture Modeling

6. Growth Mixture Modeling (GMM)
Muthén & Shedden, 1999; Muthén, 2001
Setting
A single item measured repeatedly
Hypothesized trajectory classes
Individual trajectory variation within class
Aims
Estimate trajectory shapes
Estimate trajectory class probabilities
Proportion of sample in each trajectory class
Estimate variation within class

7. Linear Growth Model Diagramx1 x2 x3 x4 x5 1 I S
mInt
mSlope
σ2Int
σ2Int,Slope 2 3 4
σ2ε1
σ2ε2
σ2ε3
σ2ε4
σ2ε5
σ2Slope
Model we saw in Nathan’s presentation. Quickly go over.

8. Linear GMM Model DiagramC
σ2Int,Slope
σ2Slope
σ2Int S I 1
mInt
mSlope 1 2 3 4 1 1 1 1 1
The change between the GMM and the previous model is the latent categorical or latent class variable, the circle with the c at the top. The circle with the c has arrows pointing toward the I and S indicating that the latent classes are based on difference in the intercept and slope of individuals – i.e. classifying individuals based on different trajectories.
Here only have I, s, but can easily have a quadratic term or non-linear growth. x1 x2 x3 x4 x5
σ2ε1
σ2ε2
σ2ε3
σ2ε4
σ2ε5

9. GMM Example Profile Plot

10. GMM example Profile Plot

11. Growth Mixture Model Equations
xitk = Interceptik + λtk*Slopeik + εitk
for individual i at time t in class k
εitk ~ N(0,σ)
General GMM with intercept and slope – can add quadratic paramater

12. LCGA Vs. GMM LCGA – Latent Class Growth Analysis
Nagin, 1999; Nagin & Tremblay, 1999
Same as GMM except no residual variance on growth factors
No individual variation within class (i.e. everyone has the same trajectory
LCGA is a special case of GMM

13. Class Enumeration Determining the number of classes Can’t use LRT Χ2
Not distributed as Χ2 due to boundary conditions (McLachlan & Peel, 2000)
Information Criteria: AIC (Akaike, 1974), BIC (Schwartz,1978)
Penalize for number of parameters and sample size
Model with lowest value
Interpretation and usefulness
Profile plot
Substantive theory
Predictive validity

14. Global Vs Local Maximum
Log Likelihood
Parameter
Global
Local
Log Likelihood
Global
Local
Why need random starts
mixtures can converge to local rather global solutions
so try multiple sets of starting values and compare the LL, parameter estimates to make sure have the global solution
Parameter

15. Open Mx Example
Take it away Ryne!

16. Selection of Mixture Genetic Analysis Writings
Growth Mixture Model
Wu et al., 2002; Kerner and Muthen, 2009; Gillespie et al., (submitted)
Latent Class Analysis
Eaves, 1993; Muthén et al., 2006; Clark, 2010
Additional References
McLachlan, Do, & Ambroise, 2004

17. Other Longitudinal Mixture models
Survival Mixture
Multiple latent classes of individuals with different survival functions
Kaplan, 2004; Masyn, 2003; Muthén & Masyn, 2005
Longitudinal Latent Class Analysis
Models patterns of change over time, rather than functional growth form
Lanza & Collins, 2006; Feldman et al., 2009
Latent Transition Analysis
Models transition from one state to another over time
Ex: Drinking alcohol or not over time
Graham et al., 1991; Nylund et al., 2006

18. Multilevel Models

19. What is Multilevel Data . . .
Most methods assume individuals are independent
Responses for one individual do not influence another individual’s responses
Multilevel, or nested data, arise when individuals are not independent
Ex: Twins in a family, students in a classroom
Share common experiences

20. . . .And why we should Care
When ignore nested structure, have underestimated standard errors
Can lead to misinterpretation of the significance of model parameters
Large body of literature about how to handle nested data
Today, focus on multilevel techniques
General multilevel texts:
Raudenbush & Bryk, 2002; Snijders & Bosker, 1999

21. Multilevel Model Equation
For individual i in cluster j:
Level One (Individual)
yij = β0j + β1j*xij + εij
Level Two (Twin Pair\Family)
β0j = γ00 + γ01*wj + μ0j
β1j = γ10 + γ11*wj + μ1j
Where εitk ~ N(0,σ), μ ~ N(0,Ψ), Cov(ε, μ) = 0
xij is an individual level covariate (age, weight)
wj is a cluster level covariate (maternal smoking)

22. Multilevel Model Equation extensions
Can have additional levels
Ex: Individuals within nuclear families with family
Can be longitudinal
Ex: Observations within individuals within families

23. Mixed Effects Vs. Multilevel Modeling
They are the same thing!!
Multilevel Model Equation:
Level One (L1):
yij = β0j + β1j*xij + εij
Level Two (L2):
β0j = γ00 + μ0j
β1j = γ10 + μ1j
Mixed Model Equation:
Plug L2 into L1, some rearranging
yij = (γ00 + μ0j) + (γ10 + μ1j) *xij + εij
yij = γ00 + γ10*xij + μ0j + μ1j*xij + εij
Fixed Effects
Random Effects

24. Multilevel Vs. Multivariate Modeling of Families
Today have dealt with multivariate analyses
Multivariate
Model for all variables for each family member
Family members can have different parameter values
Ex: different growth trajectories for parents vs. children
Only feasible when small number of family members
Ex: twins, spouses PA A C E PB

25. Multilevel Modeling of Families
Model for variation within individual and between family members
Members of a cluster are assumed statistically equivalent
i.e. Same model for each family member
Can handle various family structures
Ex: Large pedigrees, families with differing numbers of siblings
Do not have to make arbitrary assignment of family members (and checking whether assignment impacted estimates)
Ex: Assigning twins to A and B

26. Implementation of Multilevel models in open Mx
OpenMx Discussion
Discuss more tomorrow in Dynamical Systems talk

27. Multilevel Genetic Articles
General
Discuss how to extend ACDE model to twins and larger family pedigrees
Guo & Wang, 2002; McArdle & Prescott, 2005; Rabe-Hesketh, Skrondal, Gjessing, 2008
Longitudinal
McArdle, 2006
Other
Inclusion of measured genotypes: Van den Oord, 2001

29. Data Considerations Multivariate – Wide Multilevel – Long
Multiple family members per row of data
Multilevel – Long
One individual per row of data
FAMID
ZYG
ALC_T1
ALC_T2 1 20 10 2 6 15 3 4 5
FAMID
ZYG
ALC 1 20 10 2 6 15 3 4 5